We review briefly the notions of periodic frameworks and periodic deformations. Then, we propose a purely geometric criterion for characterizing auxetic one-parameter deformations.  Simply phrased, auxetic behavior means becoming laterally wider when stretched and thinner when compressed. Our geometric approach relies on the evolution of the periodicity lattice. A deformation path will be auxetic when the Gram matrix for a basis of periods gives a curve with all tangents in the positive semidefinite cone, analogous to a causal line in special relativity.  This concept is valid in arbitrary dimension. Auxetic mechanisms are then compared with expansive mechanisms defined by the stronger property that the distance between any pair of vertices increases or stays the same. For two-dimensional periodic frameworks, expansive behaviour can be explained and explored in terms of periodic pseudo-triangulations. We show how to generate this type of planar periodic frameworks and discuss their geometry. (joint work with Ileana Streinu, Smith College)